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04 Diffusion and Peer Influence (2016)

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04 Diffusion and Peer Influence (2016)

  1. 1. Diffusion and Peer Influence
  2. 2. “you are who you associate with” Time Diffusion & Peer Influence
  3. 3. 1. Diffusion A. Compartmental Models B. Network Diffusion i. Topology ii. Timing iii. Structural Transmission a. Complex contagion 2. Peer Influence
  4. 4. Coleman, Katz and Menzel, “Diffusion of an innovation among physicians” Sociometry (1957) Our substantive interest in networks is often in how things move through them, from disease to ideas to behavior. Network Diffusion & Peer Influence Basics
  5. 5. Figure 2. Binge Drinking Predicted Probabilities by Gender and Friends’ Prior Drinking. Derek A. Kreager, and Dana L. Haynie American Sociological Review 2011;76:737-763 Romantic partnerships in high school lead to adoption of partners’ friends’ behaviors Network Diffusion & Peer Influence Basics
  6. 6. Table 4. Proportion of Ties Created or Maintained Over Time by Network Process and Depression Level. David R. Schaefer et al. American Sociological Review 2011;76:764-785 Health effects on ties Depressed students form ties through non-normative network processes
  7. 7. Network Diffusion & Peer Influence Basics Classic (disease) diffusion makes use of compartmental models. Large N and homogenous mixing allows one to express spread as generalized probability models. Works very well for highly infectious bits in large populations… SI(S) model – actors are in only two states, susceptible or infectious. See: https://wiki.eclipse.org/Introduction_to_Compartment_Models for general introduction. SIIR(S) model – adds an “exposed” but not infectious state and recovered.
  8. 8. Network Diffusion & Peer Influence Basics – might even help understand the zombie apocalypse http://loe.org/images/content/091023/Zombie%20Publication.pdf
  9. 9. Network Diffusion & Peer Influence Basics Network Models Same basic SI(R,Z,etc) setup, but connectivity is not assumed random, rather it is structured by the network contact pattern. If pij is small or the network is very clustered, these two can yield very different diffusion patterns.* Real Random *these conditions do matter. Compartmental models work surprisingly well if the network is large, dense or the bit highly infectiousness…because most networks have a bit of randomness in them. We are focusing on the elements that are unique/different for network as opposed to general diffusion.
  10. 10. Network Diffusion & Peer Influence Basics If 0 < pij < 1
  11. 11. Network Diffusion & Peer Influence Basics If 0 < pij < 1 0.01 0.06 0.11 0.26 0.46
  12. 12. In addition to* the dyadic probability that one actor passes something to another (pij), two factors affect flow through a network: Topology - the shape, or form, of the network - Example: one actor cannot pass information to another unless they are either directly or indirectly connected Time - the timing of contact matters - Example: an actor cannot pass information he has not receive yet *This is a big conditional! – lots of work on how the dyadic transmission rate may differ across populations. Key Question: What features of a network contribute most to diffusion potential? Network Diffusion & Peer Influence Network diffusion features Use simulation tools to explore the relative effects of structural connectivity features
  13. 13. • A network has to be connected for a bit to pass over it • If transmission is uncertain, the longer the distance the lower the likelihood of spread. 0 0.2 0.4 2 3 4 5 6 Path distance probability Distance and diffusion (p(transfer)=pij dist Here pij of 0.6 Network Diffusion & Peer Influence Network diffusion features We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders
  14. 14. • Local clustering turns flow “in” on a potential transmission tree Arcs: 11 Largest component: 12, Clustering: 0 Arcs: 11 Largest component: 8, Clustering: 0.205 We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  15. 15. • The more alternate routes one has for transmission, the more likely flow should be. • Operationalize alternate routes with structural cohesion We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  16. 16. Probability of transfer by distance and number of non-overlapping paths, assume a constant pij of 0.6 0 0.2 0.4 0.6 0.8 1 1.2 2 3 4 5 6 Path distance probability 1 path C P X Y 10 paths 5 paths 2 paths Cohesion  Redundancy Diffusion Network Diffusion & Peer Influence Network diffusion features
  17. 17. 0 1 2 3 Node Connectivity As number of node-independent paths C P X Y Structural Cohesion: A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it. Network Diffusion & Peer Influence Network diffusion features
  18. 18. STD Transmission danger: sex or drugs? Structural core more realistic than nominal core C P X Y Data from “Project 90,” of a high-risk population in Colorado Springs Network Diffusion & Peer Influence Network diffusion features
  19. 19. • Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The assumption is that high-degree nodes are likely to contact lots of people. We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  20. 20. • Much of the work on “core groups” or “at risk” populations focus on high-degree nodes. The assumption is that high-degree nodes are likely to contact lots of people. We need: (1) reachability (2) distance (3) local clustering (4) multiple routes (5) star spreaders Network Diffusion & Peer Influence Network diffusion features
  21. 21. Network Diffusion & Peer Influence Network diffusion features Assortative mixing: A more traditional way to think about “star” effects.
  22. 22. • Simulation study: How do these different features compare over a collection of observed nets? • For each network trial: • Fix dyadic transmission probability • Randomly select a node as seed • Trace the diffusion path across the network • Measure speed & extent of spread • Model extent of spread by structural characteristics • First run: Add Health: • simple diffusion process • dyadic probability set to 0.08 • 500 trials in each network. • Then expand to : • Different assumptions of dyadic transmission probability • Different data set (Facebook) • Complex diffusion models Network Diffusion & Peer Influence Network diffusion features: simulation test
  23. 23. Network Diffusion & Peer Influence Network diffusion features: simulation test
  24. 24. Network Diffusion & Peer Influence Network diffusion features: simulation test
  25. 25. Define as a general measure of the “diffusion susceptibility” of a graph’s structure as the ratio of the area under the observed curve to the area under the curve for a matching random network. As this gets smaller than 1.0, you get effectively slower median transmission. Network Diffusion & Peer Influence Network diffusion features: simulation test
  26. 26. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  27. 27. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  28. 28. Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 1.62*** 1.90*** 1.02*** 1.81*** 1.71*** Connectivity Distance -0.207*** -0.179*** -0.171*** Independent Paths -0.077*** -0.056*** -0.052*** Distance x Paths 0.023*** 0.015*** 0.016*** Clustering Clustering Coefficient -0.692*** -0.653*** -0.454*** Grade Homophily -0.026** -0.007 -0.009* Peer Group Strength -0.868*** -0.141 -0.146 Degree Distribution Degree Skew -0.023 -0.007 -0.002 Assortative Mixing -0.189* -0.059 -0.071 Control Variables Network Size/100 0.005*** -0.005*** -.005*** 0.004* 0.002** Proportion Isolated -0.007 -1.106*** -.984*** -0.300* 0.058 Non-Complete -0.006 -0.052* -.078** -0.006 0.018 Adj- R2 0.85 0.76 0.60 0.90 0.93 N 124 124 124 124 121 Network Diffusion & Peer Influence Network diffusion features: simulation test
  29. 29. Figure 4. Relative Diffusion Ratio By Distance and Number of Independent Paths 0.4 0.6 0.8 1 1.2 2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3 Average Path Length Observed/Random k=2 k=4 k=6 k=8 Figure 4. Relative Diffusion Ratio By Distance and Number of Independent Paths 0.4 0.6 0.8 1 1.2 2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8 6.3 Average Path Length Observed/Random k=2 k=4 k=6 k=8 Network Diffusion & Peer Influence Network diffusion features: simulation test
  30. 30. Traditional “core group” models have a local-vision understanding of risk: those with lots of ties (high degree) are the focus for intervention and actions. •In the short time-windows necessary for STD transfer, low-degree networks are the relevant features for transmission. What sorts of networks emerge when average degree (in the short run) is held to small numbers? •How does the shape of the degree distribution matter? If activity is homogeneous do we get fundamentally different networks than if it is very heterogeneous? Network Diffusion & Peer Influence A closer look at emerging connectivity
  31. 31. Partner Distribution Component Size/Shape Emergent Connectivity in low-degree networks Network Diffusion & Peer Influence A closer look at emerging connectivity
  32. 32. Network Diffusion & Peer Influence A closer look at emerging connectivity
  33. 33. In both distributions, a giant component & reconnected core emerges as density increases, but at very different speeds and ultimate extent. Network Diffusion & Peer Influence A closer look at emerging connectivity
  34. 34. What distinguishes these two distributions? Shape Network Diffusion & Peer Influence A closer look at emerging connectivity
  35. 35. What distinguishes these two distributions? Shape: The scale-free network’s signature is the long-tail So what effect does changes in the shape have on connectiv Network Diffusion & Peer Influence A closer look at emerging connectivity
  36. 36.  Volume  DispersionxSkewness Network Diffusion & Peer Influence A closer look at emerging connectivity
  37. 37. Search Procedure: 1) Identify all valid degree distributions with the given mean degree and a maximum of 6 w. brute force search. 2) Map them to this space 3) Simulate networks each degree distribution 4) Measure size of components & Bicomponents Network Diffusion & Peer Influence A closer look at emerging connectivity
  38. 38. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  39. 39. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  40. 40. C:45%, B: 8.5% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  41. 41. C:45%, B: 8.5% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  42. 42. C:83%, B: 36% Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  43. 43. Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), C:99%, B: 86% Network Diffusion & Peer Influence A closer look at emerging connectivity
  44. 44. Largest Component (at least 1 path) Largest Bicomponent (at least 2 paths) Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 HD068523-01 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI), Network Diffusion & Peer Influence A closer look at emerging connectivity
  45. 45. In addition to* the dyadic probability that one actor passes something to another (pij), two factors affect flow through a network: Topology - the shape, or form, of the network - Example: one actor cannot pass information to another unless they are either directly or indirectly connected Time - the timing of contact matters - Example: an actor cannot pass information he has not receive yet *This is a big conditional! – lots of work on how the dyadic transmission rate may differ across populations. Key Question: What features of a network contribute most to diffusion potential? Network Diffusion & Peer Influence Relational Dynamics Use simulation tools to explore the relative effects of structural connectivity features
  46. 46. Three relevant networks Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: 1) The contact network. The set of pairs of people connected by sexual contact. G(V,E). 2) The exposure network. A subset of the edges in the contact network where timing makes it possible for one person to pass infection to another. 3) The transmission network. The subset of the exposure network where disease is actually passed. In most cases this is a tree layered on (2) and rooted on a source/seed node. Network Diffusion & Peer Influence Relational Dynamics
  47. 47. Contact network: Everyone, it is a connected component Who can “A” reach? Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  48. 48. Exposure network: here, node “A” could reach up to 8 others Who can “A” reach? Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  49. 49. Transmission network: upper limit is 8 through the exposure links (dark blue). Transmission is path dependent: if no transmission to B, then also none to {K,L,O,J,M} Who can “A” reach? Exposable Link (from A’s p.o.v.) Contact Network Diffusion & Peer Influence Relational Dynamics Discussions of network effects on STD spread often speak loosely of “the network.” There are three relevant networks that are often conflated: Three relevant networks
  50. 50. The mapping between the contact network and the exposure network is based on relational timing. In a dynamic network, edge timing determines if something can flow down a path because things can only be passed forward in time. Definitions: Two edges are adjacent if they share a node. A path is a sequence of adjacent edges (E1, E2, …Ed). A time-ordered path is a sequence of adjacent edges where, for each pair of edges in the sequence, the start time Si is less than or equal to Ej S(E1) < E(E2) Adjacent edges are concurrent if they share a node and have start and end dates that overlap. This occurs if: S(E2) < E(E1) Concurrency Network Diffusion & Peer Influence Relational Dynamics
  51. 51. A B C D time 1 2 3 4 5 6 7 8 9 10 AB BC CE E CD 2 - 71 - 3 S(ab) E(ab) S(bc) E(bc) S(ce) E(ce) The mapping between the contact network and the exposure network is based on relational timing. In a dynamic network, edge timing determines if something can flow down a path because things can only be passed forward in time. Concurrency Network Diffusion & Peer Influence Relational Dynamics
  52. 52. The constraints of time-ordered paths change our understanding of the system structure of the network. Paths make a network a system: linking actors together through indirect connections. Relational timing changes how paths cumulate in networks. Indirect connectivity is no longer transitive: A B C D1 - 2 3 - 4 1 - 2 Here A can reach C, and C and reach D. But A cannot reach D (nor D A). Why? Because any infection A passes to C would have happened after the relation between C and D ended. A B C D1 - 2 3 - 4 1 - 2 Network Diffusion & Peer Influence Relational Dynamics
  53. 53. Edge time structures are characterized by sequence, duration and overlap. Paths between i and j, have length and duration, but these need not be symmetric even if the constituent edges are symmetric. Network Diffusion & Peer Influence Relational Dynamics
  54. 54. 1 2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 2 1 1 1 2 2 2 2 2 1 Implied Contact Network of 8 people in a ring All relations Concurrent Reachability = 1.0 Network Diffusion & Peer Influence Relational Dynamics
  55. 55. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Implied Contact Network of 8 people in a ring Serial Monogamy (1) 1 2 3 7 6 5 8 4 Reachability = 0.71 Network Diffusion & Peer Influence Relational Dynamics
  56. 56. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Implied Contact Network of 8 people in a ring Mixed Concurrent 2 2 1 1 2 2 3 3 Reachability = 0.57 Network Diffusion & Peer Influence Relational Dynamics
  57. 57. Implied Contact Network of 8 people in a ring Serial Monogamy (3) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 2 Reachability = 0.43 Network Diffusion & Peer Influence Relational Dynamics
  58. 58. 1 2 1 1 2 1 2 2 Timing alone can change mean reachability from 1.0 when all ties are concurrent to 0.42. In general, ignoring time order is equivalent to assuming all relations occur simultaneously – assumes perfect concurrency across all relations. Network Diffusion & Peer Influence Relational Dynamics
  59. 59. A B C D E F A 0 1 2 2 4 1 B 1 0 1 2 3 2 C 0 1 0 1 2 2 D 0 0 1 0 1 1 E 0 0 0 1 0 2 F 1 0 0 1 0 0 While a is 2 steps from d, and d is 1 step from e, a and e are 4 steps apart. This is because the shorter path from a to e emerges after the path from d to e ended. 4 2 1 Path distances no longer simply add Network Diffusion & Peer Influence Relational Dynamics
  60. 60. A B C D 3 - 4 E The geodesic from A to D is AE, ED and is two steps long. But the fastest path would be AB, BC, CD, which while 3 steps long could get there by time 5 compared to time 7. I ignore this feature for the remainder… Aside: “shortest” can now be replaced with “fastest” Network Diffusion & Peer Influence Relational Dynamics
  61. 61. Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: All edges concurrent a b c d e f g h i j k l m a . 1 1 1 1 1 1 1 1 1 1 1 1 b 1 . 1 1 1 1 1 1 1 1 1 1 1 c 1 1 . 1 1 1 1 1 1 1 1 1 1 d 1 1 1 . 1 1 1 1 1 1 1 1 1 e 1 1 1 1 . 1 1 1 1 1 1 1 1 f 1 1 1 1 1 . 1 1 1 1 1 1 1 g 1 1 1 1 1 1 . 1 1 1 1 1 1 h 1 1 1 1 1 1 1 . 1 1 1 1 1 i 1 1 1 1 1 1 1 1 . 1 1 1 1 j 1 1 1 1 1 1 1 1 1 . 1 1 1 k 1 1 1 1 1 1 1 1 1 1 . 1 1 l 1 1 1 1 1 1 1 1 1 1 1 . 1 m 1 1 1 1 1 1 1 1 1 1 1 1 . Network Diffusion & Peer Influence Relational Dynamics
  62. 62. All edges except gd concurrent a b c d e f g h i j k l m a . 1 1 1 1 1 1 1 1 1 1 1 1 b 1 . 1 1 1 1 1 1 1 1 1 1 1 c 1 1 . 1 1 1 1 1 1 1 1 1 1 d 1 1 1 . 1 1 1 1 1 1 1 1 1 e 1 1 1 1 . 1 1 1 1 1 1 1 1 f 1 1 1 1 1 . 1 1 1 1 1 1 1 g 0 0 0 1 0 0 . 1 1 1 1 1 1 h 0 0 0 1 0 0 1 . 1 1 1 1 1 i 0 0 0 1 0 0 1 1 . 1 1 1 1 j 0 0 0 1 0 0 1 1 1 . 1 1 1 k 0 0 0 1 0 0 1 1 1 1 . 1 1 l 0 0 0 1 0 0 1 1 1 1 1 . 1 m 0 0 0 1 0 0 1 1 1 1 1 1 . Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: Network Diffusion & Peer Influence Relational Dynamics
  63. 63. Concurrency affects exposure by making paths symmetric, which increases exposure down multiple “branches” of a contact sequence. Consider a simplified example: All edges gd after all others: a b c d e f g h i j k l m a . 1 1 1 1 1 1 0 0 0 0 0 0 b 1 . 1 1 1 1 1 0 0 0 0 0 0 c 1 1 . 1 1 1 1 0 0 0 0 0 0 d 1 1 1 . 1 1 1 0 0 0 0 0 0 e 1 1 1 1 . 1 1 0 0 0 0 0 0 f 1 1 1 1 1 . 1 0 0 0 0 0 0 g 0 0 0 1 0 0 . 1 1 1 1 1 1 h 0 0 0 0 0 0 1 . 1 1 1 1 1 i 0 0 0 0 0 0 1 1 . 1 1 1 1 j 0 0 0 0 0 0 1 1 1 . 1 1 1 k 0 0 0 0 0 0 1 1 1 1 . 1 1 l 0 0 0 0 0 0 1 1 1 1 1 . 1 m 0 0 0 0 0 0 1 1 1 1 1 1 . The concurrency status of {dg} determines which “side” of the graph is exposed. Note this effect happens at the system level – the correlation between exposure and node-level timing is essentially zero (d/g excepted) Network Diffusion & Peer Influence Relational Dynamics
  64. 64. Resulting infection trace from a simulation (Morris et al, AJPH 2010). Observed infection paths from 10 seeds in an STD simulation, edges coded for concurrency status. Network Diffusion & Peer Influence Relational Dynamics
  65. 65. Resulting infection trace from a simulation (Morris et al, AJPH 2010). Network Diffusion & Peer Influence Relational Dynamics Observed infection paths from 10 seeds in an STD simulation, edges coded for concurrency status.
  66. 66. Timing constrains potential diffusion paths in networks, since bits can flow through edges that have ended. This means that: • Structural paths are not equivalent to the diffusion-relevant path set. • Network distances don’t build on each other. • Weakly connected components overlap without diffusion reaching across sets. • Small changes in edge timing can have dramatic effects on overall diffusion • Diffusion potential is maximized when edges are concurrent and minimized when they are “inter-woven” to limit reachability. Combined, this means that many of our standard path-based network measures will be incorrect on dynamic graphs. Network Diffusion & Peer Influence Relational Dynamics
  67. 67. The distribution of paths is important for many of the measures we typically construct on networks, and these will be change if timing is taken into consideration: Centrality: Closeness centrality Path Centrality Information Centrality Betweenness centrality Network Topography Clustering Path Distance Groups & Roles: Correspondence between degree-based position and reach-based position Structural Cohesion & Embeddedness Opportunities for Time-based block-models (similar reachability profiles) In general, any measures that take the systems nature of the graph into account will differ. Structural measurement implications Network Diffusion & Peer Influence Relational Dynamics – other implication of dynamic nets (briefly)
  68. 68. New versions of classic reachability measures: 1) Temporal reach: The ij cell = 1 if i can reach j through time. 2) Temporal geodesic: The ij cell equals the number of steps in the shortest path linking i to j over time. 3) Temporal paths: The ij cell equals the number of time-ordered paths linking i to j. These will only equal the standard versions when all ties are concurrent. Duration explicit measures 4) Quickest path: The ij cell equals the shortest time within which i could reach j. 5) Earliest path: The ij cell equals the real-clock time when i could first reach j. 6) Latest path: The ij cell equals the real-clock time when i could last reach j. 7) Exposure duration: The ij cell equals the longest (shortest) interval of time over which i could transfer a good to j. Each of these also imply different types of “betweenness” roles for nodes or edges, such as a “limiting time” edge, which would be the edge whose comparatively short duration places the greatest limits on other paths. Structural measurement implications Network Diffusion & Peer Influence Relational Dynamics – other implication of dynamic nets (briefly)
  69. 69. Topology & Time interact: How relational sequencing affects diffusion is conditioned by the structural patterns of relations. Examples: - Time limitations mean star nodes can’t interact with everyone at each time tick; effects of high degree are thus limited by schedule/availabiltiy - If within-cluster ties are also more frequent than between cluster ties, then the effects of communities will be magnified. -Multiple connectivity should provide routes around breaks built by temporal sequence. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  70. 70. Measures Dependent variable: Reachability in the exposure graph. This is the proportion of pairs in the network that are reachable in time. Exposure Graph Density Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  71. 71. Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. 1 2 3 4 5 6 7 8 9 1 -- 1 1 1 1 1 1 1 1 2 1 -- 2 2 2 1 1 1 1 3 1 2 -- 2 2 1 1 1 1 4 1 2 2 -- 2 1 1 1 1 5 1 2 2 2 -- 1 1 1 1 6 1 1 1 1 1 -- 2 2 1 7 1 1 1 1 1 2 -- 2 1 8 1 1 1 1 1 2 2 -- 1 9 1 1 1 1 1 1 1 1 -- Cell Value = highest k-connected component pair belongs to. Average = 1.25 Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  72. 72. 1 2 3 4 5 6 7 8 9 1 -- 1 1 1 1 1 1 1 1 2 1 -- 3 3 3 2 2 2 1 3 1 3 -- 3 3 2 2 2 1 4 1 3 3 -- 3 2 2 2 1 5 1 3 3 3 -- 2 2 2 1 6 1 2 2 2 2 -- 2 2 1 7 1 2 2 2 2 2 -- 2 1 8 1 2 2 2 2 2 2 -- 1 9 1 1 1 1 1 1 1 1 -- Cell Value = highest k-connected component pair belongs to. Average = 1.6 Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  73. 73. Measures: Independent variables Features of the topology. Of key interest is the level of structural cohesion. Average connectivity Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  74. 74. Volume Distance Connectivity Nodes: 148 Mean Deg: 6.16 Density: 0.042 Centralization: 0.187 Nodes: 80 Mean Deg: 5.27 Density: 0.067 Centralization: 0.373 Nodes: 154 Mean Deg: 3.71 Density: 0.025 Centralization: 0.147 Nodes: 128 Mean Deg: 3.39 Density: 0.027 Centralization: 0.205 Mean: 3.59 Diameter: 5 Centralization: 0.312 Mean: 3.02 Diameter: 5 Centralization: 0.413 Mean: 4.99 Diameter: 8 Centralization: 0.259 Mean: 4.55 Diameter: 6 Centralization: 0.301 Largest BC: 0.51 Pairwise K: 1.57 Largest BC: 0.33 Pairwise K: 1.34 Largest BC: 0.08 Pairwise K: 1.07 Largest BC: Pairwise K: 1.06 Exemplar independent variables “HighCohesive”“LowCohesive”Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  75. 75. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  76. 76. “Low Cohesive” Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  77. 77. Proportion of relations concurrent DensityoftheExposureNetwork Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  78. 78. Colors=different nets Panels=cohesion level Proportion of relations concurrent DensityoftheExposureNetwork Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  79. 79. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  80. 80. 1. Concurrency has a necessarily positive effect on potential diffusion exposure 1. This implies that we should see greater transmission given greater concurrency 2. This works by creating “multiple routes” in the exposure path structure 2. Structural cohesion captures multiple routes in the contact graph 1. Higher levels of cohesion increase exposure by directly increasing the underlying transmission substrate 3. There is a negative interaction between cohesion and concurrency: as cohesion increases, the relative returns to concurrency decrease. 1. But this comes at the cost of a higher base-level of exposure. Network Diffusion & Peer Influence Structural Moderators of Timing Effects
  81. 81. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Thus far we have focused on a “simple” dyadic diffusion parameter, pij, where the probability of passing/receiving the bit is purely dependent on discordant status of the dyad, sometimes called the “independent cascade model” (), which suggests a monotonic relation between the number of times you are exposed through peers. High exposure could be due to repeated interaction with one person or weak interaction with many, effectively equating: Alternative models exist. Under “complex contagion” for example, the likelihood that I accept the bit that flows through the network depends on the proportion of my peers that have the bit.
  82. 82. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion 1 1 2 3 Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters:
  83. 83. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: Assume pij=1, k=2, starting nodes in yellow
  84. 84. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: For this network under weak complex diffusion (k=2), the maximum risk size is 8.
  85. 85. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Assume adoption requires k neighbors having adopted, then transmission can only occur within dense clusters: For this network under weak complex diffusion (k=2), the maximum risk size is reaches 98%. One of the Prosper schools: Start
  86. 86. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Can lead to widely varying sizes of potential diffusion cascades. Here’s the distribution across all PROPSPER schools: Distribution is largely bimodal (even with a connected pair start)
  87. 87. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Can lead to widely varying sizes of potential diffusion cascades. Here’s the distribution across all PROPSPER schools: The governing factors are (a) curved effect of local redundancy and (b) structural cohesion Network Average Proportion Reached k=2 complex contagion MeanCascadeSize Coh=0.3 Coh=1.2 Coh=2.2 Coh=3.2 Coh=4.1
  88. 88. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex Contagion Does get used for real health work: Here , authors assume a CC process, seeded with observed depressive cases, turn that into a Markov model and ask what parameters would maximize fit from simulated to observed.
  89. 89. Network Diffusion & Peer Influence Structural Transmission Dynamics: beyond disease diffusion Complex diffusion is just the most well studied of the options that combine transmission with some pairwise positional feature. This is a wide-open area for future research. The basic idea is that transmission is increased/decreased if there is some third structural property that the susceptible & infected pair share. This leads us into the general problem of peer influence models…when do peers change each other’s behaviors?
  90. 90. Background: • Long standing research interest in how our relations shape our attitudes and behaviors. • Most often assumed mechanism is that people (through conversation or similar) change each others beliefs/opinions, which changes behavior. This implies that position in a communication network should be related to attitudes. • Alternatives: • Modeling behavior: ego copies behavior of alter to gain respect, esteem, etc. • Distinction: Ego tries to be different from (some) alter to gain respect, esteem, etc. • Access: Ego wants to do Y, but can only do so because alter provides access (say, being old enough to buy cigarettes). Network Diffusion & Peer Influence Peer Influence Dynamics
  91. 91. Background: • Early work was ego-centric – people informed on their peers •Seems to have inflated PI effects by ~50% or so…either through projection of ego behavior onto peers or selective interaction (what alters do with ego may be different than what alter does all the time). •Then to cross sectional associations based on alter self-reports •Better, but still likely conflates selection with influence •Next to dynamic models: •Ego Behavior(t) ~ f(ego behavior(t-1) + alter behavior (t-1) + controls •Much better; still debate on (a) correct estimation functions, (b) unobserved selection features that confound causal inference. •Development of Actor-oriented models (SIENA) Network Diffusion & Peer Influence Peer Influence Dynamics
  92. 92. Background: •Finally: Experimental manipulation of peer exposure •“Gold standard” for isolation of peer effects •Likely strongly underestimates effects (as measure intent to treat, not take- up of treatment, since people may not care about relations that can be manipulated). b(Peer(y)): Ego Inform < Alter Inform < Cross Sectional < Dynamic < Experimental. Still often find peer effects, but my sense is that we’ve (strongly) over-corrected at this point. Network Diffusion & Peer Influence Peer Influence Dynamics
  93. 93. Freidkin’s Structural Theory of Social Influence : Two-part model: Beliefs are a function of two sources: a) Individual characteristics •Gender, Age, Race, Education, Etc. Standard sociology b) Interpersonal influences •Actors negotiate with others Network Diffusion & Peer Influence Peer Influence Dynamics
  94. 94. XBY )1( (1) )1()1()( )1( YWYY αα Tt   (2) Y(1) = an N x M matrix of initial opinions on M issues for N actors X = an N x K matrix of K exogenous variable that affect Y B = a K x M matrix of coefficients relating X to Y a = a weight of the strength of endogenous interpersonal influences W = an N x N matrix of interpersonal influences Network Diffusion & Peer Influence Peer Influence Dynamics
  95. 95. XBY )1( (1) This is the standard sociology model for explaining anything: the General Linear Model. It says that a dependent variable (Y) is some function (B) of a set of independent variables (X). At the individual level, the model says that:  k kiki BXY Usually, one of the X variables is e, the model error term. Network Diffusion & Peer Influence Peer Influence Dynamics
  96. 96. )1()1()( )1( YWYY αα Tt   (2) This part of the model taps social influence. It says that each person’s final opinion is a weighted average of their own initial opinions )1( )1( Yα And the opinions of those they communicate with (which can include their own current opinions) )1( T αWY Network Diffusion & Peer Influence Peer Influence Dynamics
  97. 97. The key to the peer influence part of the model is W, a matrix of interpersonal weights. W is a function of the communication structure of the network, and is usually a transformation of the adjacency matrix. In general:    j ij ij w w 1 10 Various specifications of the model change the value of wii, the extent to which one weighs their own current opinion and the relative weight of alters. Network Diffusion & Peer Influence Peer Influence Dynamics
  98. 98. 1 2 3 4 1 2 3 4 1 1 1 1 0 2 1 1 1 0 3 1 1 1 1 4 0 0 1 1 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 1 2 3 4 1 .50 .25 .25 0 2 .25 .50 .25 0 3 .20 .20 .40 .20 4 0 0 .33 .67 Even 2*self 1 2 3 4 1 .50 .25 .25 0 2 .25 .50 .25 0 3 .17 .17 .50 .17 4 0 0 .50 .50 degree Self weight: 1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 2 1 4 0 0 1 2 1 2 3 4 1 2 1 1 0 2 1 2 1 0 3 1 1 3 1 4 0 0 1 1 Network Diffusion & Peer Influence Peer Influence Dynamics
  99. 99. )1()1()( )1( YWYY αα Tt   Formal Properties of the model When interpersonal influence is complete, model reduces to: )1( )1()1()( 01     T Tt WY YWYY When interpersonal influence is absent, model reduces to: )1( )1()1()( 0 Y YWYY   Tt (2) Network Diffusion & Peer Influence Peer Influence Dynamics
  100. 100. Formal Properties of the model The model is directly related to spatial econometric models: If we allow the model to run over t and W remains constant: XBWYY )1()()( αα   eb   XWYY ~)()( α Where the two coefficients (a and b) are estimated directly (See Doreian, 1982, SMR). This is the linear network auto correlation model, best bet with cross-sectional data (and randomization trick to estimate se) Network Diffusion & Peer Influence Peer Influence Dynamics
  101. 101. Simple example 1 2 3 4 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 Y 1 3 5 7 a = .8 T: 0 1 2 3 4 5 6 7 1.00 2.60 2.81 2.93 2.98 3.00 3.01 3.01 3.00 3.00 3.21 3.33 3.38 3.40 3.41 3.41 5.00 4.20 4.20 4.16 4.14 4.14 4.13 4.13 7.00 6.20 5.56 5.30 5.18 5.13 5.11 5.10 By t=7, still variability in Y Network Diffusion & Peer Influence Peer Influence Dynamics
  102. 102. 1 2 3 4 1 2 3 4 1 .33 .33 .33 0 2 .33 .33 .33 0 3 .25 .25 .25 .25 4 0 0 .50 .50 Y 1 3 5 7 a = 1.0 1.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81 3.00 3.00 3.33 3.56 3.68 3.74 3.78 3.81 5.00 4.00 4.00 3.92 3.88 3.86 3.85 3.84 7.00 6.00 5.00 4.50 4.21 4.05 3.95 3.90 By t=7, almost no variability in Y T: 0 1 2 3 4 5 6 7 Simple example Network Diffusion & Peer Influence Peer Influence Dynamics
  103. 103. Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: Network Diffusion & Peer Influence Peer Influence Dynamics
  104. 104. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  105. 105. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  106. 106. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  107. 107. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  108. 108. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  109. 109. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  110. 110. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  111. 111. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations
  112. 112. Extended example: building intuition Consider a network with three cohesive groups, and an initially random distribution of opinions: Now weight in-group ties higher than between group ties Network Diffusion & Peer Influence Peer Influence Dynamics
  113. 113. Simulated Peer Influence: 75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2
  114. 114. Consider the implications for populations of different structures. For example, we might have two groups, a large orthodox population and a small heterodox population. We can imagine the groups mixing in various levels: Little Mixing Moderate Mixing Heavy Mixing .95 .05 .05 .02 .95 .008 .008 .02 .95 .001 .001 .02 Heterodox: 10 people Orthodox: 100 People Network Diffusion & Peer Influence Peer Influence Dynamics
  115. 115. Light Heavy Moderate
  116. 116. Light mixing
  117. 117. Light mixing
  118. 118. Light mixing
  119. 119. Light mixing
  120. 120. Light mixing
  121. 121. Light mixing
  122. 122. Moderate mixing
  123. 123. Moderate mixing
  124. 124. Moderate mixing
  125. 125. Moderate mixing
  126. 126. Moderate mixing
  127. 127. Moderate mixing
  128. 128. High mixing
  129. 129. High mixing
  130. 130. High mixing
  131. 131. High mixing
  132. 132. High mixing
  133. 133. High mixing
  134. 134. In an unbalanced situation (small group vs large group) the extent of contact can easily overwhelm the small group. Applications of this idea are evident in: •Missionary work (Must be certain to send missionaries out into the world with strong in-group contacts) •Overcoming deviant culture (I.e. youth gangs vs. adults) •This is also the mechanism behind why most youth peer influence is a *good* thing – most youth are well behavior and civic minded…so are exerting positive influences on their peers. Network Diffusion & Peer Influence Peer Influence Dynamics
  135. 135. Friedkin (1998) generalizes the model so that alpha varies across people. (1) simply changing a to a vector (A), which then changes each person’s opinion directly (2) by linking the self weight (wii) to alpha. )1()1()( )( YAIAWYY  Tt Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A further restriction on the model sets wii = 1-aii This leads to a great deal more flexibility in the theory, and some interesting insights. Consider the case of group opinion leaders with unchanging opinions (I.e. many people have high aii, while a few have low): Network Diffusion & Peer Influence Peer Influence Dynamics
  136. 136. Group 1 Leaders Group 2 Leaders Group 3 Leaders Peer Opinion Leaders
  137. 137. Peer Opinion Leaders
  138. 138. Peer Opinion Leaders
  139. 139. Peer Opinion Leaders
  140. 140. Peer Opinion Leaders
  141. 141. Peer Opinion Leaders
  142. 142. Further extensions of the model might: • Time dependent a: people likely value other’s opinions more early than later in a decision context • Interact a with XB: people’s self weights are a function of their behaviors & attributes • Make W dependent on structure of the network (weight transitive ties greater than intransitive ties, for example) • Time dependent W: The network of contacts does not remain constant, but is dynamic, meaning that influence likely moves unevenly through the network • And others likely abound…. Network Diffusion & Peer Influence Peer Influence Dynamics
  143. 143. There are two common ways to test for peer associations through networks. The first estimates the parameters (a and b) of the network autocorrelation model directly, the second transforms the network into a dyadic model, predicting similarity among actors. eb   XWYY ~)()( α Peer influence model: Network Diffusion & Peer Influence Peer Influence Dynamics This is the linear network autocorrelation model, and as specified, the model makes strong assumptions about equilibrium opinion and static relations.  Some variants on this also expand e to include alternative autocorrelation in the error structure.
  144. 144. There are two common ways to test for peer associations through networks. The first estimates the parameters (a and b) of the network autocorrelation model directly, the second transforms the network into a dyadic model, predicting similarity among actors. eb   XWYY ~)()( α Peer influence model: Network Diffusion & Peer Influence Peer Influence Dynamics Note that since WY is a a simple vector -- weighted mean of friends Y -- which can be constructed and added to your GLM model. That is, multiple Y by a W matrix, and run the regression with WY as a new variable, and the regression coefficient is an estimate of a. This is what Doriean calls the QAD estimate of peer influence. It’s wrong, a will be biased, but it’s often not terribly wrong if most obvious selection factors are built int0 X
  145. 145. An obvious problem with this specification is that cases are, by definition, not independent, hence “network autocorrelation” terminology. In practice, the QAD approach (perhaps combined with a GLS estimator) results in empirical estimates that are “virtually indistinguishable” from MLE (Doreian et al, 1984) The proper way to estimate the peer equation is to use maximum likelihood estimates, and Doreian gives the formulas for this in his paper, and Carter Butts has implemented in in R with the LNAM procedure. An alternative is to use non-parametric approaches, such as the Quadratic Assignment Procedure, to estimate the effects. Network Diffusion & Peer Influence Peer Influence Dynamics
  146. 146. Peer influence through Dyad Models Another way to get at peer influence is not through the level of Y, but by assessing the similarity of connected peers. Recall the simulated example: peer influence is reflected in how close points are to each other. Network Diffusion & Peer Influence Peer Influence Dynamics
  147. 147. Peer influence through Dyad Models The model is now expressed at the dyad level as: ij k kkijij eXbAbbY  10 Where Y is a matrix of similarities, A is an adjacency matrix, and Xk is a matrix of similarities on attributes Advantages include ease of specifying relation-specific similarity functions. You can add different features of a relation by adjusting/adding “Aij” variables. Disadvantage is that now in addition to network autocorrelation, you have repeated cases (on both sides). But these can be dealt with using non-parametric modeling & testing techniques (QAP, for example). (which we will go over this afternoon) Network Diffusion & Peer Influence Peer Influence Dynamics
  148. 148. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  149. 149. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  150. 150. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  151. 151. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  152. 152. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  153. 153. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  154. 154. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  155. 155. Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  156. 156. The network shows significant evidence of weight-homophily Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  157. 157. Effects of peer obesity on ego, by peer type Edge-wise regressions of the form: ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb Ego is repeated for all alters; models include random effects on ego id Used the friend/relative tracking data from a larger heart-health study to identify network contacts, including friends. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  158. 158. This modeling strategy pools observations on edges and estimates a global effect net of change in ego/alter as a control. Here color is a single ego, number is wave (only 2 egos and 3 waves represented). Effects of peer obesity on ego, by peer type ControlsEgoAltAltEgo previouspreviousCurrentCurrent  )()()( 321 bbb 1 Ego-Current Alter Current 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 Peer Effect Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  159. 159. Alterative specifications include using change- change models and allowing for a random effect of peers. This allows for greater variability in peer effects, and the potential to model differences. ee currentpreviouseCurrentprevious ControlstAltAlEgoEgo bb b   )()( 1 Ego-Current Alter Current 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 b be1 be1 Effects of peer obesity on ego, by peer type Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies ee previouspreviousCurrenteCurrent ControlsEgoAltAltEgo bb bbb   )()()( 32 Or difference models:
  160. 160. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F The C&F studies – of obesity, but also other work on the FHS data – turn on the validity of the causal association. All turn on some issue of model miss-specification, typically: • Can’t truly distinguish a network effect from other sources of common influence • “Selection” (“homophily”) or “Common influence” (“Shared environment”) • The most strident work in this area (Salizi • Statistical errors • Misinterpretation of confidence intervals • Poorly specified/estimated models C&H do a nice job of laying out their responses here: http://jhfowler.ucsd.edu/examining_dynamic_social_networks.pdf and here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2597062/
  161. 161. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Cohen-Cole, E. and Fletcher, J. M. (2008). Detecting implausible social network effects in acne, height, and headaches: longitudinal analysis. British Medical Journal 337 a2533. Use the same models as C&F on Add Health to show that things which are theoretically unlikely to be contagious appear to be in this form of model. Note these coefficients are substantially smaller than C&F and only significant at the 0.1 level; and not robust to any sensitivity analysis.
  162. 162. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 1) C&F claim that differences in directional effects support a PI story: • C& F: While mutual friends and egoalter friends are > 0, alterego is not, means ego is emulating alter. • Lyons notes these CIs overlap too much to make any claim about distinguishing them from each other.
  163. 163. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 2) Insufficient controls for Homophily • C& F: Use of alter’s lagged Y to control for homophily. Logic is that any feature that selected us to be friends at t-1 would have had it’s effect then. • Lyons notes that current and lagged have opposite signs, which seems suspect, and anyway is an insufficient control. He’s likely right here… 3) Directionality cannot distinguish the source of association • C& F: the ordering: mutual, egoalter, alterego suggests an “esteem” model, where ego copies the behavior of alter. • Lyons argues that we would expect the same logic from a simple “foci” of similarity. I don’t find this argument convincing. 4) Random permutation tests cannot establish 3-degree rule • C& F: Association between alters at 1, 2, 3 degrees of separation are higher than we’d expect by chance, based on a permutation test. • Lyons invalid if the data are incomplete, which they certainly are. I don’t find this argument convincing…data are always incomplete…
  164. 164. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. 5) The models are statistically inconsistent (if not incoherent) • C& F: Use separate models for each type of tie, with random effects on ego. • Lyons notes that these really should be treated as simultaneous equations, with shared error structures and so forth. Doing so (a) leads to unidentified models that must force the estimation of the peer effect to 0. That observed ^0 indicates something amiss. • Strikes me as a bit down in the weeds and I’m not convinced here that he’s critiquing them for what they are really doing (argues there are more equations than data, which is patently not true).
  165. 165. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Critiques of C&F Lyons, 2011. My sense is that the strategy C&F took was not fundamentally misguided, but the model specification is probably thin; certainly in the obesity paper – less so in some of the later papers appearing after these debates. Ideally you’d have a much better direct model for selection – perhaps even a separate two-stage model (see the Siena module), but here there are very limited observational controls, which would have been easy to add. In later specifications, they do add fixed effects for ego and still find similar results. Commenting on the debate on SocNET – and a related conclusion that only experiments could provide valid inference – Tom Snijders says: “The logical consequence of this is that we are stuck with imperfect methods. Lyons argues as though only perfect methods are acceptable, and while applauding such lofty ideals I still believe that we should accept imperfection, in life as in science. Progress is made by discussion and improvement of imperfections, not by their eradication.” For a full general discussion, see : https://www.lists.ufl.edu/cgi-bin/wa?A2=ind1106&L=SOCNET&P=R11428
  166. 166. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded So long as there is an unobserved X that causes both ties and behavior, the effect of peers is unidentified.
  167. 167. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded Only route out is to make X fully informed (or informing) by an observable Z….but realistically there are few things that (a) cause behavior exclusively without any selection pressure (a) or cause ties exclusively without any influence pressure (b) (though note b is what experimental assignments do) (X causes Z, not Y directly) (X causes A, not Y directly)
  168. 168. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Shalizi & Thomas: PI is *generally* confounded Should be noted that this is true for *any* effect – there’s always the potential that an unobserved latent variable is creating a spurious effect; This sort of work argues that the only solution is to use experimental (or, sometimes, propensity score style models)…but that’s simply not always feasible practically. We need to beware of making the best the enemy of the good enough…lest we make no progress at all…
  169. 169. Willard Van Quine, professor of philosophy and mathematics emeritus from Harvard University who is regarded as one of the most famous philosophers in the world, wrote his doctoral thesis on a 1927 Remington typewriter, which he still uses. However, he "had an operation on it" to change a few keys to accommodate special symbols. "I found I could do without the second period, the second comma -- and the question mark.” "You don't miss the question mark?” "Well, you see, I deal in certainties." Selection or Influence? Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  170. 170. Is it all selection •What do we know about how friendships form? •Opportunity / focal factors - Being members of the same group - In the same class - On the same team - Members of the same church •Structural Relationship factors - Reciprocity - Social Balance •Behavior Homophily - Smoking - Drinking Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  171. 171. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Network Model Coefficients, In school Networks Network Diffusion & Peer Influence
  172. 172. How to correct this problem? •Essentially, this is an omitted variable problem, and my “solution” has been to identify as many potentially relevant alternative variables as I can find. •The strongest possible correction is to use fixed-effects* models that control for all non-varying individual covariates. These have their own problems… •Dual model for influence & selection. •Two-stage model “Heckman” sorts of models •Dynamic SAOM models Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies *“Adding fixed effects to dynamic panel models with many subjects and few repeat observations creates severe bias towards zero coefficients. This has been demonstrated both analytically (Nickell 1981) and through simulations (Nerlove 1971) for OLS and other regression models and has been well-known by social scientists, including economists, for a very long time. In fact, CCF even note that they do not add fixed effects to their logit regression model for this reason, but they strangely assert that fixed effects are necessary in the OLS model.” Estimating Peer Effects on Health in Social Networks : A Response to Cohen-Cole and Fletcher; Trogdon, Nonnemaker, Pais J.H. Fowler, PhD and N.A. Christakis, MD, PhD
  173. 173. • Causal status of such similarity is hard to know, • Identification strategies are stringent • My sense is we’re over-correcting on this front; let’s figure out what’s there first. Selection Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Y X1 X2 Weak instruments bias us toward null effects Y X1 X2 I
  174. 174. Possible solutions: • Theory: Given what we know about how friendships form, is it reasonable to assume a bi-directional cause? That is, work through the meeting, socializing, etc. process and ask whether it makes sense that Y is a cause of W. This will not convince a skeptical reader, but you should do it anyway. • Models: - Time Order. Necessary but not sufficient. We are on somewhat firmer ground if W precedes Y in time, but the Shalizi & Thomas problem of an as-yet-earlier joint confounder is still there. - Simultaneous Models. Model both the friendship pattern and the outcome of interest simultaneously. Best bet for direct estimation •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  175. 175. Table 4. Selected SIENA Parameter Estimates: Parental Knowledge, Parental Discipline, and Drinking a Model 2 b SE t SD Selection parameters Alter effects: Who is more often named as a friend? Parental knowledge -0.002 0.004 -0.47 0.003 Parental discipline -0.004 0.002 -1.55 0.001 Drinking 0.083 0.010 8.69 *** 0.007 Ego effects: Who names more friends? Parental knowledge 0.044 0.007 5.85 *** 0.039 Parental discipline -0.002 0.005 -0.36 0.023 Drinking -0.011 0.021 -0.53 0.089 Similarity effects: Choosing friends similar to oneself Parental knowledge 0.169 0.025 6.70 *** 0.101 Parental discipline 0.151 0.017 8.86 *** 0.035 Drinking 0.276 0.021 13.45 *** 0.006 Behavioral parameters: Influence on Drinking Friends' attributes Mean Parental knowledge -0.230 0.065 -3.56 ** 0.014 Mean Parental discipline -0.051 0.043 -1.18 0.011 Drinking mean similarity 1.162 0.110 10.56 *** 0.023 Control variables (individual level) Parental knowledge -0.122 0.014 -8.93 *** 0.004 Parental discipline -0.043 0.009 -4.54 *** 0.002 ***p < .001. **p < .01. *p < .05. †p < .10. a Models also include rate and shape parameters, structural parameters, and the full set of alter, ego, similarity, and individual-level control parameters SIENA model of drinking Daniel T. Ragan, D. Wayne Osgood
  176. 176. Possible solutions: •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies
  177. 177. Possible solutions: •Sensitivity Analysis: I think the most reasonable solution…take error potential seriously, attempt to evaluate how big a problem it really is. Network Diffusion & Peer Influence Peer Influence & Health: Current Lit & Controversies Sociological Methods & Research 2000

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